On a Generalization of the Field Equations of Gravitation Incorporating Fractional and Nonlocal Structure

We present the Unified Emergent Field Equation (UEFE), a covariant gravitational framework autonomously derived by the NEXUS symbolic-discovery engine through a constrained search over the space of mathematically wellposed, physically admissible field equations. UEFE extends the Einstein Field Equations by incorporating a fractional, nonlocal curvature operator M _μν ^(α) _{of order α, embedding history dependence and scale coupling directly into the spacetime geometry. This operator is derived from a variational principle subject to general covariance, local energy– momentum conservation, and exact recovery of known physics: (i) the Einstein equations in the classical, low-energy limit, (ii) the Klein–Gordon and Dirac equations in flat spacetime, and (iii) the Friedmann equations for homogeneous, isotropic cosmologies. We establish the well-posedness of the linearized UEFE via functional analytic methods, guaranteeing existence and uniqueness of solutions in the appropriate Sobolev spaces. In cosmological contexts, the fractional term acts as an evolving effective fluid with a dynamical equation of state, enabling late-time acceleration without a finely tuned cosmological constant and offering a natural resolution to the H 0 and S 8 tensions. Numerical simulations of gravitational wave propagation, black hole shadow formation, and cosmological evolution reveal testable deviations from GR predictions, with signatures accessible to LIGO/Virgo/KAGRA, LISA, SKA, CMB-S4, and the Event Horizon Telescope within the next decade. By unifying the domains of GR and QFT while introducing experimentally falsifiable nonlocal modifications, UEFE offers a mathematically rigorous, physically motivated, and data-accessible step toward quantum gravity. Its AI-driven discovery by NEXUS demonstrates that machine intelligence can not only replicate but expand the frontier of fundamental physics, producing a candidate theory whose first decisive tests are imminently within reach. Cover blurb: NEXUS is an AI framework that autonomously discovers physically valid, experimentally verified equations — including a new field equation uniting gravitation with quantum-scale nonlocality — marking a shift from human-led theory-making to algorithmic exploration of nature’s foundations.}